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\title{复变函数练习4.3 - 解析函数的泰勒展开 }
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\date{2024 年 5 月 13 日}
%\date{March 9, 2021}

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\begin{document}

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\begin{enumerate}

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\item  %Problem 01
（泰勒定理）设 $f(z)$ 在区域 $D$ 内解析，设 $a\in D$, 设圆 $K:|z-a|<R$ 含于 $D$, 则 $f(z)$ 在 $K$ 内能展开成唯一的幂级数 
$f(z)=\sum\limits_{n=0}^{\infty} c_n(z-a)^n$, 其中系数 $c_n$ 可由 $f(z)$ 的积分或微分得到，
$$c_n = \frac{1}{2\pi i} \int_{\Gamma_\rho} \frac{f(\xi)d\xi }{(\xi-a)^{n+1}} = \frac{f^{(n)}(a)}{n!}. $$
其中 $\Gamma_\rho$ 为圆周 $|\xi-a|=\rho<R$. 
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\item  %Problem 02
证明函数 $f(z)$ 在区域 $D$ 内解析的充分必要条件是 $f(z)$ 在 $D$ 内任意一点 $a$ 都可以展开成 $z-a$ 的幂级数。

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\item  %Problem 03
设幂级数 $f(z)=\sum\limits_{n=0}^{\infty} c_n(z-a)^n$ 的收敛半径是 $R>0$. 
证明 $f(z)$ 在圆周 $C: |z-a|=R$ 上至少有一个奇点。

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\item  %Problem 04
求函数 $f(z)=e^z$ 在 $z=0$ 处的泰勒展开。

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\item  %Problem 05
求多值函数 $\mathrm{Ln}(z)$ 的各支在 $z=0$ 处的泰勒展开。

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\item  %Problem 06
设 $\alpha$ 是一个复数。求多值函数 $(1+z)^\alpha$ 的各支在 $z=0$ 处的泰勒展开。


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\item  %Problem 07
求函数 $f(z)=\frac{e^z}{1-z}$ 在 $z=0$ 处的泰勒展开。
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\item  %Problem 08
求函数 $\sqrt{z+i}$ 在 $z=0$ 处的泰勒展开。

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\item  %Problem 08
求函数 $e^z\cos(z)$ 与 $e^z\sin(z)$ 在 $z=0$ 处的泰勒展开。

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\item  %Problem 09
求函数 $f(z)=\frac{z}{z+2}$ 在 $z=1$ 处的泰勒展开。

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\item  %Problem 10
设有下述泰勒展开，(1) 证明 $c_n=c_{n-1}+c_{n-2}(n\ge 2)$; (2) 求出展式的前五项；(3)求出收敛范围。
$$ \frac{1}{1-z-z^2} = \sum\limits_{n=0}^{\infty} c_nz^n. $$
 
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\item  %Problem 11
将下述函数展成 $z$ 的幂级数，并指出展式成立的范围， 
$$
(1) f(z)=\int_0^z\exp(z^2)dz; \hspace{1cm} 
(2) f(z)=\int_0^z \frac{\sin z}{z} dz; \hspace{1cm} 
(3) f(z)=\sin^2z. 
$$

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\end{enumerate}


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\end{document}

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